A variety is a class of algebras in the same language that is
closed under taking direct products, subalgebras, and homomorphic
images. By Tarski's Theorem
([1]}),
the smallest variety containing a class
\({\mathcal {K}}\)  of algebras in the same
language equals \(\mathbb {H}\mathbb {S}\mathbb {P}{\mathcal {K}}\) .
Birkhoff's Theorem
([1]}) characterizes varieties as
classes of algebras defined by identities.
In more detail, an identity in a
language \(F\)  is an expression
of the form
\(s(x_1,\dots ,x_n)=t(x_1,\dots ,x_n)\)  where \(s(x_1,\dots ,x_n)\)  and
\(t(x_1,\dots ,x_n)\)  are terms in
the language \(F\) .
An algebra \(\mathbf {A}\)  in the
language \(F\)  satisfies an identity
\(s(x_1,\dots ,x_n)=t(x_1,\dots ,x_n)\) 
if the \(n\) -ary term operations \(s^{\mathbf {A}}\)  and \(t^{\mathbf {A}}\)  are equal.
Birkhoff's Theorem is the statement that for a fixed language,
a class \({\mathcal {V}}\)  of algebras
is a variety if and only if there exists a set \(\Sigma \)  of identities
such that an algebra lies in \({\mathcal {V}}\)  if and only if
it satisfies the identities in \(\Sigma \) .
